Analysis

all_task<- cbind(s2r_abs,s2v_abs) %>% 
  select(sid, frame_effect_r, frame_effect_v, frame_size)


p_df<- perception_abs %>%
  ungroup() %>% 
  select(magnitude, sid) %>% 
  rename(sub_1 =sid)

all_task<- cbind(all_task,p_df)

all_task<- all_task %>% 
  select(-sub_1)

all_task<-all_task %>% 
  rename(vv= frame_effect_v,
         oc = frame_effect_r,
         perc = magnitude)

across_frames<- perc_x_sacc_df %>%
  pivot_wider(names_from = task, values_from = magnitude)

Perception Task: RFI

To determine the change in PSE as a function of frame size, we subtracted the PSE for counterclockwise trials from clockwise trials, and then divided that by half.

A negative value indicates that participants PSEs are being biased in the opposite direction of the tilt of the frame.

(#tab:percept summary stat table)
Perception Task
Frame Size Mean Median SD Min Max
Small 2.11 2.00 1.15 -0.27 6.35
Medium 1.91 1.71 1.12 0.15 6.68
Large 1.71 1.45 1.02 -0.06 4.77
Extra Large 1.55 1.33 0.99 -0.50 5.80

Perception Task: RFI

frame_size

estimate

statistic

p

parameter

Method

Alternative

95% CI

175

2.11

16.44

< .001***

79.00

One Sample t-test

two.sided

[1.85, 2.37]

410

1.91

15.27

< .001***

79.00

One Sample t-test

two.sided

[1.66, 2.16]

645

1.71

14.97

< .001***

79.00

One Sample t-test

two.sided

[1.48, 1.93]

880

1.55

14.03

< .001***

79.00

One Sample t-test

two.sided

[1.33, 1.77]

p_mag_anova<- aov(frame_effect_r ~FRAME_SIZE_VAL, data =saccade_to_rod_magnitude)
summary(p_mag_anova)
##                 Df Sum Sq Mean Sq F value  Pr(>F)    
## FRAME_SIZE_VAL   3   46.9  15.635   9.284 6.7e-06 ***
## Residuals      316  532.2   1.684                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tukey_p<-tukey_hsd(p_mag_anova)
tukey_p
## # A tibble: 6 × 9
##   term           group1 group2 null.value estimate conf.low conf.high     p.adj
## * <chr>          <chr>  <chr>       <dbl>    <dbl>    <dbl>     <dbl>     <dbl>
## 1 FRAME_SIZE_VAL 175    410             0    0.730    0.200     1.26  0.00242  
## 2 FRAME_SIZE_VAL 175    645             0    0.992    0.462     1.52  0.0000123
## 3 FRAME_SIZE_VAL 175    880             0    0.851    0.321     1.38  0.000251 
## 4 FRAME_SIZE_VAL 410    645             0    0.262   -0.268     0.792 0.578    
## 5 FRAME_SIZE_VAL 410    880             0    0.121   -0.409     0.651 0.935    
## 6 FRAME_SIZE_VAL 645    880             0   -0.141   -0.671     0.389 0.902    
## # ℹ 1 more variable: p.adj.signif <chr>
apa_table(tukey_p)
(#tab:perc aov frame size effect)
**
term group1 group2 null.value estimate conf.low conf.high p.adj p.adj.signif
FRAME_SIZE_VAL 175 410 0.00 0.73 0.20 1.26 0.00 **
FRAME_SIZE_VAL 175 645 0.00 0.99 0.46 1.52 0.00 ****
FRAME_SIZE_VAL 175 880 0.00 0.85 0.32 1.38 0.00 ***
FRAME_SIZE_VAL 410 645 0.00 0.26 -0.27 0.79 0.58 ns
FRAME_SIZE_VAL 410 880 0.00 0.12 -0.41 0.65 0.94 ns
FRAME_SIZE_VAL 645 880 0.00 -0.14 -0.67 0.39 0.90 ns

The results between the table and graph are inconsistent due to different tests being used. I am unsure whether I need to use the Tukey HSD test ( shown in the table above) or if a Holm-Bonferroni correction is more appropriate (used in the violin graph below).

Saccade-to-Vertical Task: Visuovestibular Effect

The effect of the frames was quantified by subtracting the mean errors for the counterclockwise-tilted frames from those of the clockwise-tilted frames then halving this value to get a measure of the average effect of a single frame (negative values indicated eye movements that deviated in the direction opposite the tilt of the frame).

(#tab:s2v summary stat table)
Saccade-to-Vertical Task
Frame Size Mean Median SD Min Max
Small 1.47 1.31 1.63 -2.77 8.74
Medium 1.23 1.01 1.75 -1.84 8.34
Large 1.14 1.04 1.35 -1.63 5.91
Extra Large 1.04 0.88 1.51 -2.10 8.97 
s2v_ttest_df<-saccade_to_vert_magnitude %>% group_by(FRAME_SIZE_VAL) %>% do(tidy(t.test(.$frame_effect_v)))
nice_table(s2v_ttest_df)

FRAME_SIZE_VAL

estimate

statistic

p

parameter

Method

Alternative

95% CI

175

1.47

8.04

< .001***

79.00

One Sample t-test

two.sided

[1.11, 1.83]

410

1.23

6.29

< .001***

79.00

One Sample t-test

two.sided

[0.84, 1.62]

645

1.14

7.55

< .001***

79.00

One Sample t-test

two.sided

[0.84, 1.44]

880

1.04

6.16

< .001***

79.00

One Sample t-test

two.sided

[0.70, 1.37]

vv_mag_anova<- aov(frame_effect_v~FRAME_SIZE_VAL, data =saccade_to_vert_magnitude)
#summary(vv_mag_anova)
tukey_vv<-tukey_hsd(vv_mag_anova)
apa_table(tukey_vv)
(#tab:s2v aov frame size effect)
**
term group1 group2 null.value estimate conf.low conf.high p.adj p.adj.signif
FRAME_SIZE_VAL 175 410 0.00 -0.24 -0.88 0.40 0.77 ns
FRAME_SIZE_VAL 175 645 0.00 -0.33 -0.97 0.31 0.56 ns
FRAME_SIZE_VAL 175 880 0.00 -0.43 -1.07 0.21 0.31 ns
FRAME_SIZE_VAL 410 645 0.00 -0.09 -0.73 0.55 0.98 ns
FRAME_SIZE_VAL 410 880 0.00 -0.19 -0.83 0.45 0.87 ns
FRAME_SIZE_VAL 645 880 0.00 -0.10 -0.75 0.54 0.98 ns 

## `geom_smooth()` using formula = 'y ~ x'

Visuvestibular models predicting perceptual effect for each frame size

Small frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4004 -0.7165 -0.1283  0.4315  4.2230 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.12753    0.17419  12.214   <2e-16 ***
## vv          -0.01199    0.07954  -0.151    0.881    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.155 on 78 degrees of freedom
## Multiple R-squared:  0.0002913,  Adjusted R-squared:  -0.01253 
## F-statistic: 0.02273 on 1 and 78 DF,  p-value: 0.8805

Medium frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7622 -0.6009 -0.1230  0.4030  3.2499 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.57513    0.13952  11.290  < 2e-16 ***
## vv           0.27040    0.06549   4.129 9.06e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.019 on 78 degrees of freedom
## Multiple R-squared:  0.1794, Adjusted R-squared:  0.1689 
## F-statistic: 17.05 on 1 and 78 DF,  p-value: 9.057e-05

Large frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8761 -0.6915 -0.1754  0.5270  3.5536 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.37309    0.13883   9.890 2.05e-15 ***
## vv           0.29264    0.07861   3.723 0.000371 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9464 on 78 degrees of freedom
## Multiple R-squared:  0.1509, Adjusted R-squared:   0.14 
## F-statistic: 13.86 on 1 and 78 DF,  p-value: 0.0003706

Extral large frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_xl)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.55711 -0.62980 -0.02815  0.50862  2.08057 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.18155    0.11387  10.376 2.40e-16 ***
## vv           0.35265    0.06248   5.644 2.57e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.837 on 78 degrees of freedom
## Multiple R-squared:   0.29,  Adjusted R-squared:  0.2809 
## F-statistic: 31.86 on 1 and 78 DF,  p-value: 2.569e-07

Saccade to Rod : Orientation Contrast Effect

The effect of the frames was quantified by subtracting the mean errors for the counterclockwise-tilted frames from those of the clockwise-tilted frames then halving this value to get a measure of the average effect of a single frame (negative values indicated eye movements that deviated in the direction opposite the tilt of the frame).

(#tab:s2r summary stat table)
Saccade-to-Rod Task: Orientation Contrast Effect
Frame Size Mean Median SD Min Max
Small -1.61 -1.50 1.44 -6.47 4.51
Medium -0.88 -0.99 1.24 -4.38 2.39
Large -0.62 -0.58 1.35 -4.01 4.02
Extra Large -0.76 -0.70 1.14 -4.76 2.57

FRAME_SIZE_VAL

estimate

statistic

p

parameter

Method

Alternative

95% CI

175

-1.61

-9.98

< .001***

79.00

One Sample t-test

two.sided

[-1.93, -1.29]

410

-0.88

-6.36

< .001***

79.00

One Sample t-test

two.sided

[-1.15, -0.60]

645

-0.62

-4.09

< .001***

79.00

One Sample t-test

two.sided

[-0.92, -0.32]

880

-0.76

-5.94

< .001***

79.00

One Sample t-test

two.sided

[-1.01, -0.50]

##                 Df Sum Sq Mean Sq F value  Pr(>F)    
## FRAME_SIZE_VAL   3   46.9  15.635   9.284 6.7e-06 ***
## Residuals      316  532.2   1.684                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(#tab:s2r within graph)
**
term group1 group2 null.value estimate conf.low conf.high p.adj p.adj.signif
FRAME_SIZE_VAL Small Medium 0.00 -0.73 -1.26 -0.20 0.00 **
FRAME_SIZE_VAL Small Large 0.00 -0.99 -1.52 -0.46 0.00 ****
FRAME_SIZE_VAL Small Extra Large 0.00 -0.85 -1.38 -0.32 0.00 ***
FRAME_SIZE_VAL Medium Large 0.00 -0.26 -0.79 0.27 0.58 ns
FRAME_SIZE_VAL Medium Extra Large 0.00 -0.12 -0.65 0.41 0.94 ns
FRAME_SIZE_VAL Large Extra Large 0.00 0.14 -0.39 0.67 0.90 ns

Combined Saccades and Perception Task Comparison

So far the perceptual and orientation contrast effect were reported as negative numbers, indicating that the perceptual response or saccade erred in the opposite direction of the tilt of the frame. However, it should be noted that for the purpose of making an additive comparison between summed saccade tasks and the perceptual response, we used the inverse value of the OC effect.

## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between across_frames$perception and
## across_frames$combined_saccade is positive, statistically significant, and very
## large (r = 0.43, 95% CI [0.33, 0.51], t(318) = 8.45, p < .001)
## `geom_smooth()` using formula = 'y ~ x'

## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_175$perception and comb_sacc_perc_175$combined_saccade
## t = 1.2689, df = 78, p-value = 0.2083
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.08000506  0.35096232
## sample estimates:
##       cor 
## 0.1422117
## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_410$perception and comb_sacc_perc_410$combined_saccade
## t = 4.9876, df = 78, p-value = 3.619e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3049679 0.6420802
## sample estimates:
##       cor 
## 0.4917352
## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_645$perception and comb_sacc_perc_645$combined_saccade
## t = 4.9228, df = 78, p-value = 4.662e-06
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.2991548 0.6383051
## sample estimates:
##     cor 
## 0.48687
## 
##  Pearson's product-moment correlation
## 
## data:  comb_sacc_perc_880$perception and comb_sacc_perc_880$combined_saccade
## t = 5.7131, df = 78, p-value = 1.935e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.3672545 0.6815287
## sample estimates:
##      cor 
## 0.543144
## Effect sizes were labelled following Funder's (2019) recommendations.
## 
## The Pearson's product-moment correlation between across_frames$perception and
## across_frames$combined_saccade is positive, statistically significant, and very
## large (r = 0.43, 95% CI [0.33, 0.51], t(318) = 8.45, p < .001)
## `geom_smooth()` using formula = 'y ~ x'

Perception and Saccade-to-Vertical

## `geom_smooth()` using formula = 'y ~ x'

Perception and Saccade-to-Rod

## `geom_smooth()` using formula = 'y ~ x'

## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_175$perception and s2r_perc_175$s2r
## t = -2.1634, df = 78, p-value = 0.03357
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.43490358 -0.01920867
## sample estimates:
##        cor 
## -0.2379214
## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_410$perception and s2r_perc_410$s2r
## t = -2.6151, df = 78, p-value = 0.0107
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.47405942 -0.06846902
## sample estimates:
##        cor 
## -0.2839148
## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_645$perception and s2r_perc_645$s2r
## t = -3.4645, df = 78, p-value = 0.0008661
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.5414599 -0.1581624
## sample estimates:
##        cor 
## -0.3651887
## 
##  Pearson's product-moment correlation
## 
## data:  s2r_perc_880$perception and s2r_perc_880$s2r
## t = -2.6239, df = 78, p-value = 0.01045
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.47480242 -0.06942329
## sample estimates:
##        cor 
## -0.2847961

Task by Frame Size

#two way anova task X frame size summary stats

task_by_frame_long %>% 
  group_by(frame_size,task) %>% 
  get_summary_stats(magnitude,type = "mean_sd")
## # A tibble: 8 × 6
##   frame_size task  variable      n  mean    sd
##   <fct>      <fct> <fct>     <dbl> <dbl> <dbl>
## 1 175        vv    magnitude    80 1.47   1.63
## 2 175        oc    magnitude    80 1.61   1.44
## 3 410        vv    magnitude    80 1.23   1.75
## 4 410        oc    magnitude    80 0.88   1.24
## 5 645        vv    magnitude    80 1.14   1.35
## 6 645        oc    magnitude    80 0.618  1.35
## 7 880        vv    magnitude    80 1.04   1.51
## 8 880        oc    magnitude    80 0.759  1.14
#box plot 
task_frame_bxp<- ggboxplot(
  task_by_frame_long, x = "frame_size", y = "magnitude", color = "task", pallet ="jco")

task_frame_bxp

Model Comparison

For each frame size a hierarchical design was employed using two models: 1) model 1 predicted the overall RFI magnitude (measured by the perception task) from the visuovestibular effect (measured by the saccade-to-vertical task) and 2) model 2 predicted the overall RFI magnitude from visuovestibular effect and the orientation contrast effect (measured by the saccade-to-rod task).

Small Frame

Perceptual effect predicted by visuovestibular effect

## 
## Call:
## lm(formula = perc ~ vv, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4004 -0.7165 -0.1283  0.4315  4.2230 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.12753    0.17419  12.214   <2e-16 ***
## vv          -0.01199    0.07954  -0.151    0.881    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.155 on 78 degrees of freedom
## Multiple R-squared:  0.0002913,  Adjusted R-squared:  -0.01253 
## F-statistic: 0.02273 on 1 and 78 DF,  p-value: 0.8805

Perceptual effect predicted by orientation contrast effect

oc_small<- lm(perc~ oc, df_small)
summary(oc_small)
## 
## Call:
## lm(formula = perc ~ oc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4518 -0.6510 -0.2152  0.3203  4.3107 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.80516    0.18865   9.569 8.55e-15 ***
## oc           0.18931    0.08751   2.163   0.0336 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.122 on 78 degrees of freedom
## Multiple R-squared:  0.05661,    Adjusted R-squared:  0.04451 
## F-statistic:  4.68 on 1 and 78 DF,  p-value: 0.03357

Model comparison

## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_small)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.4789 -0.6643 -0.1876  0.3187  4.2826 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.83034    0.21902   8.357 2.08e-12 ***
## vv          -0.01792    0.07779  -0.230   0.8184    
## oc           0.19003    0.08810   2.157   0.0341 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.129 on 77 degrees of freedom
## Multiple R-squared:  0.05726,    Adjusted R-squared:  0.03277 
## F-statistic: 2.338 on 2 and 77 DF,  p-value: 0.1033
## [1] 0.05696534
## Analysis of Variance Table
## 
## Model 1: perc ~ vv
## Model 2: perc ~ vv + oc
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1     78 104.12                              
## 2     77  98.19  1    5.9331 4.6527 0.03412 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tinytable_qokepuft57llugx3c4ol
Model 1 Model 2
(Intercept) 2.128 1.830
(0.174) (0.219)
vv -0.012 -0.018
(0.080) (0.078)
oc 0.190
(0.088)
Num.Obs. 80 80
R2 0.000 0.057
R2 Adj. -0.013 0.033
AIC 254.1 251.4
BIC 261.3 260.9
Log.Lik. -124.057 -121.710
RMSE 1.14 1.11

Medium Frame

summary(model_1_medium)
## 
## Call:
## lm(formula = perc ~ vv, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7622 -0.6009 -0.1230  0.4030  3.2499 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.57513    0.13952  11.290  < 2e-16 ***
## vv           0.27040    0.06549   4.129 9.06e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.019 on 78 degrees of freedom
## Multiple R-squared:  0.1794, Adjusted R-squared:  0.1689 
## F-statistic: 17.05 on 1 and 78 DF,  p-value: 9.057e-05
## 
## Call:
## lm(formula = perc ~ oc, data = df_medium)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.5927 -0.6444 -0.2625  0.3698  4.1965 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.68232    0.14824  11.349   <2e-16 ***
## oc           0.25647    0.09807   2.615   0.0107 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.078 on 78 degrees of freedom
## Multiple R-squared:  0.08061,    Adjusted R-squared:  0.06882 
## F-statistic: 6.839 on 1 and 78 DF,  p-value: 0.0107
## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_medium)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.92902 -0.56106 -0.08099  0.47815  3.13629 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.39053    0.15335   9.068 8.87e-14 ***
## vv           0.25768    0.06352   4.056 0.000118 ***
## oc           0.22763    0.08988   2.533 0.013356 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.985 on 77 degrees of freedom
## Multiple R-squared:  0.2425, Adjusted R-squared:  0.2228 
## F-statistic: 12.32 on 2 and 77 DF,  p-value: 2.273e-05
## [1] 0.0631005
## Analysis of Variance Table
## 
## Model 1: perc ~ vv
## Model 2: perc ~ vv + oc
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1     78 80.926                              
## 2     77 74.703  1    6.2227 6.4141 0.01336 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tinytable_xlrswibxkgbtyto2hvgv
Model 1 Model 2
(Intercept) 1.575 1.391
(0.140) (0.153)
vv 0.270 0.258
(0.065) (0.064)
oc 0.228
(0.090)
Num.Obs. 80 80
R2 0.179 0.242
R2 Adj. 0.169 0.223
AIC 234.0 229.5
BIC 241.1 239.1
Log.Lik. -113.975 -110.775
RMSE 1.01 0.97

Large Frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8761 -0.6915 -0.1754  0.5270  3.5536 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.37309    0.13883   9.890 2.05e-15 ***
## vv           0.29264    0.07861   3.723 0.000371 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9464 on 78 degrees of freedom
## Multiple R-squared:  0.1509, Adjusted R-squared:   0.14 
## F-statistic: 13.86 on 1 and 78 DF,  p-value: 0.0003706
## 
## Call:
## lm(formula = perc ~ oc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.5026 -0.7240 -0.3182  0.5357  3.4002 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.53706    0.11769  13.060  < 2e-16 ***
## oc           0.27628    0.07974   3.465 0.000866 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9561 on 78 degrees of freedom
## Multiple R-squared:  0.1334, Adjusted R-squared:  0.1223 
## F-statistic:    12 on 1 and 78 DF,  p-value: 0.0008661
## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_large)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6401 -0.5876 -0.2332  0.2946  3.9705 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.28410    0.13581   9.455 1.59e-14 ***
## vv           0.24787    0.07650   3.240  0.00177 ** 
## oc           0.22699    0.07681   2.955  0.00415 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9027 on 77 degrees of freedom
## Multiple R-squared:  0.2373, Adjusted R-squared:  0.2175 
## F-statistic: 11.98 on 2 and 77 DF,  p-value: 2.948e-05
## [1] 0.08649254
## Analysis of Variance Table
## 
## Model 1: perc ~ vv
## Model 2: perc ~ vv + oc
##   Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
## 1     78 69.863                                
## 2     77 62.747  1    7.1162 8.7326 0.004146 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tinytable_csjdf6zboqcj9tb76eim
Model 1 Model 2
(Intercept) 1.373 1.284
(0.139) (0.136)
vv 0.293 0.248
(0.079) (0.076)
oc 0.227
(0.077)
Num.Obs. 80 80
R2 0.151 0.237
R2 Adj. 0.140 0.218
AIC 222.2 215.6
BIC 229.3 225.1
Log.Lik. -108.096 -103.798
RMSE 0.93 0.89

Extra Large Frame

## 
## Call:
## lm(formula = perc ~ vv, data = df_xl)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.55711 -0.62980 -0.02815  0.50862  2.08057 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.18155    0.11387  10.376 2.40e-16 ***
## vv           0.35265    0.06248   5.644 2.57e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.837 on 78 degrees of freedom
## Multiple R-squared:   0.29,  Adjusted R-squared:  0.2809 
## F-statistic: 31.86 on 1 and 78 DF,  p-value: 2.569e-07
oc_xl<- lm(perc~ oc, df_xl)
summary(oc_xl)
## 
## Call:
## lm(formula = perc ~ oc, data = df_xl)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.7607 -0.6438 -0.1995  0.5181  3.7649 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.36114    0.12804  10.630   <2e-16 ***
## oc           0.24608    0.09378   2.624   0.0105 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9522 on 78 degrees of freedom
## Multiple R-squared:  0.08111,    Adjusted R-squared:  0.06933 
## F-statistic: 6.885 on 1 and 78 DF,  p-value: 0.01045
## 
## Call:
## lm(formula = perc ~ vv + oc, data = df_xl)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.73611 -0.57671  0.00541  0.53652  2.24999 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.09815    0.12229   8.980 1.31e-13 ***
## vv           0.32691    0.06345   5.152 1.93e-06 ***
## oc           0.14516    0.08372   1.734   0.0869 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8264 on 77 degrees of freedom
## Multiple R-squared:  0.3167, Adjusted R-squared:  0.2989 
## F-statistic: 17.84 on 2 and 77 DF,  p-value: 4.294e-07
## [1] 0.02668074
## Analysis of Variance Table
## 
## Model 1: perc ~ vv
## Model 2: perc ~ vv + oc
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1     78 54.640                              
## 2     77 52.586  1    2.0533 3.0066 0.08693 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
tinytable_3jahd2r2rsmemnws5opn
Model 1 Model 2
(Intercept) 1.182 1.098
(0.114) (0.122)
vv 0.353 0.327
(0.062) (0.063)
oc 0.145
(0.084)
Num.Obs. 80 80
R2 0.290 0.317
R2 Adj. 0.281 0.299
AIC 202.5 201.5
BIC 209.7 211.0
Log.Lik. -98.264 -96.732
RMSE 0.83 0.81